In the last century or so a large number of techniques have been developed to make (normally invisible) phase variations visible in imaging devices. These techniques include Zernike phase contrast, Nomarski differential interference contrast, generalized phase contrast, Foucault knife edge, shadowgraph and dark-field.
More recently there has been progress in extending some of these techniques to X-ray imaging. Phase-contrast X-ray imaging offers the ability to image soft tissues, where conventional absorption X-ray imaging does not produce a detectible contrast due to low absorption in soft tissues. A number of new phase-contrast techniques have been developed to address the particularly difficult nature of X-rays. These difficulties relate primarily to the difficulty of focusing and imaging X-ray beams, due to both their penetrating power and the small range of refractive indices available. Current techniques include TIE (Transport of Intensity Equation), phase contrast imaging and ptychography. Another current technique is known as “X-ray Talbot interferometry” (XTI), which yields intermediate images that encode one or more differential phase images. The XTI method when implemented using simple linear gratings gives one differential phase image. XTI implemented with two dimensional (crossed) gratings yields an x differential phase image and a y differential phase image. The differential phase in x and y directions will be referred to as ‘phase’ hereafter in this disclosure for simplicity.
A problem with phase imaging is that the recovered phase information is almost always ‘wrapped’ because it is embedded in a sinusoidal function. In other words, the phase value recovered from the phase imaging system in question is the underlying phase value modulo 2π radians. A typical phase unwrapping method looks for discontinuities in the phase and compensates for sudden jumps or drops by adding or subtracting integer multiples of 2π to restore the true phase values.
Phase unwrapping is in general a difficult problem because the problem is inherently ambiguous and there is no consistent way to distinguish between discontinuities caused by wrapping, and discontinuities which exist in the true, unwrapped, phase. A phase discontinuity is associated with a “residue”, which is a single point of discontinuity in the phase that constitutes the start or end-point of an extended region of discontinuity.
To unwrap the phase, some methods operate in three steps, namely finding residues, joining residues together to minimize some measure of discontinuity, and then using a flood-fill algorithm to correct the phase values in regions.
However, because these algorithms are region-based, when they fail they can produce extremely visible artifacts, such as an offset of 2π which covers a substantial fraction of the area of an image.
FIG. 3 illustrates an example of prior art incorrect phase unwrapping results. FIG. 3 is a hypothetical one-dimensional example of a genuine phase edge treated as a phase wrapping effect by a prior art phase unwrapping algorithm. FIG. 3 depicts a real discontinuity in a phase signal, depicted in (310), in which a signal 311 gradually drops to a minimum value 312 and then jumps to a much higher value 313 before it slowly decreases again as depicted at 314. One current phase unwrapping method will detect this discontinuity and shift the region 314 on the right of the discontinuity 312/313 downward by 2π based on the fact that slopes of the signal just before (see 316) and after (see 315) the discontinuity point 312/133 seem to match. However, the result depicted at 320 not only mistakenly removes a real phase edge, but it also offsets the phase value by 2π, an error that will propagates to the end of the signal.